This is a comment which won't fit in the comment box.

Let me see if I understand the question. There are three equalities: $$\dim \ H^0(X, L_{\lambda}) = \int_X e^{\lambda+\rho} \prod \frac{\alpha}{e^{\alpha/2} - e^{-\alpha/2}} \quad (1)$$ $$\int_X e^{\lambda+\rho} \prod \frac{\alpha}{e^{\alpha/2} - e^{-\alpha/2}} = \left. \left( \sum_{w \in W} e^{w(\lambda+\rho)} {\LARGE /} \prod(e^{\alpha/2} - e^{- \alpha/2}) \right)\right|_{0} \quad (2)$$ $$ \left. \left( \sum_{w \in W} e^{w(\lambda+\rho)} {\LARGE /} \prod(e^{\alpha/2} - e^{- \alpha/2}) \right)\right|_{0} = \prod \frac{\langle \lambda+\rho, \alpha \rangle}{\langle \rho, \alpha \rangle} \quad (3)$$

Here $|_{0}$ means that the formula, being a power series in the weights of $T$, is a function on the Cartan algebra $\mathfrak{h}$, and we are to evaluate this function at the origin.

Is your question how to go from the LHS of $(2)$ to the RHS of $(3)$ without passing through the RHS of $(2)$?